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I was thinking of a problem with entry and exit of agents. So say that n agents of each of the two types enter every period. Types change in the sense that after a type 1 trades with a type 2, he becomes a new type (someone with the same preferences, but a new "endowment"), but types do not change randomly over time. An agent can choose to leave the game whenever he/she wants. (This question is based on work by Gale and McClennan and Sonnenschien.)

Is there any way to rule out an equilibrium which discriminates against type 2? If yes, how?